Introduction

Partizan End Nim is a game played by two players called Left and Right. Initially there are n stacks of boxes in a row, each stack containing at least one box. Players take turns reducing the number of boxes from the stack on their respective side (Left removes from the leftmost stack, while Right removes from the rightmost stack). For example, the position 3 | 5 | 2 (or, denoted more tersely, 352) has three boxes in its leftmost pile, five boxes in the middle pile, and two boxes in the rightmost pile. When the game starts, Left can only remove boxes from the pile of size three, and Right can only remove boxes from the pile of size two. The first player that cannot move loses. This particular position should be a win for the first player (whether that be Left or Right). The first player should remove a whole pile, for the stack of size 5 dominates the remaining stack. Notice that if one player has a legal move from a position, the other player can also legally move, making the game all small, as defined in and.

In our version of Partizan End Nim, piles are not limited to finite size. A move in Partizan End Nim requires the player to change the size of the closest pile to a smaller ordinal number (possibly 0).